MinT - Best Known (119−44, 119, s)-Nets in Base 9

Best Known (119−44, 119, s)-Nets in Base 9

Digital (75, 119, 448)-net over F₉, using

5 times m-reduction [i] based on digital (75, 124, 448)-net over F₉, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

Digital (75, 119, 945)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹⁹, 945, F₉, 44) (dual of [945, 826, 45]-code), using
- 825 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0) [i] based on linear OA(9⁴⁴, 45, F₉, 44) (dual of [45, 1, 45]-code or 45-arc in PG(43,9)), using
  - dual of repetition code with length 45 [i]

There is no (75, 119, 164178)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 358850 292805 854303 523098 629990 015854 912822 862007 016488 275788 755607 640422 696755 719501 076480 055643 288078 723705 258913 > 9¹¹⁹ [i]